Your best bet is to form an LLC in Delaware, Nevada or Wyoming. Each state has attractive terms & conditions for businesses registered there. One quality that all three share is anonymity of business owners. If you form an LLC then you would be considered a disregarded entity to the IRS. This would allow you to avoid applying for an employer identification number (EIN), which would require your name in the application.
The attorneys on our site –– have helped many business owners retain their anonymity when forming their businesses. Check us out and message me with any comments on your experience (I’m the CEO!). Good luck!